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Full text of "Men Of Mathematics"

of assuming that people are depraved because they drink to
excess, Galton inverted this hypothesis and assumed temporarily
that people drink to excess because they have inherited no moral
guts from their ancestors, in short, because they are depraved.
Brushing aside all the vaporous moralizing of the reformers,
Galton took a firm grip on a scientific, unemotional, workable
hypothesis to which he could apply the impartial machinery of
mathematics. His work has not yet registered socially. For the
moment we need note only that Galton, like Abel, inverted his
problem - turned it upside-down and inside-out, back-end-to
and foremost-end-backward. Like Hiawatha and his fabulous
mittens, Galton put the skinside inside and the inside outside.
All this is far from being obvious or a triviality. It is one of
the most powerful methods of mathematical discovery (or
invention) ever devised, and Abel was the first human being to
use it consciously as an engine of research. 'You must always
invert', as Jacobi said when asked the secret of his mathema-
tical discoveries. He was recalling what Abel and he had done*
If the solution of a problem becomes hopelessly involved, try
turning the problem backwards, put the quaesita for the data
and vice versa. Thus if we find Cardan's character incompre-
hensible when we think of him as a son of his father, shift the
emphasis, invert it, and see what we get when we analyse
Cardan's father as the begetter and endower of his son. Instead
of studying 'inheritance' concentrate on 'endowing'. To return
to those who remember some trigonometry.
Suppose mathematicians had been so blind as not to see that
sin #, cos # and the other direct trigonometric functions are
simpler to use, in the addition formulae and elsewhere, than the
inverse functions sin-1 a, cos-1 02. Recall the formula sin (x + y)
in terms of sines and cosines of x and y, and contrast it with the
formula for sin"1 (x + y) in terms of # and y. Is not the former
incomparably simpler, more elegant, more 'natural' than the
latter? Now, in the integral calculus, the inverse trigonometric
functions present themselves naturally as definite integrals'of
simple algebraic irrationalities (second degree); such integrals
appear when we seek to find the length of an arc of a circle by
means of the integral calculus. Suppose the inverse trigono-